New Cases of Solow-sustainability with Exhaustible Resources

New Cases of Solow-sustainability with Exhaustible Resources John M. Hartwick Economics, Queen s University, March 204 March 6, 204 Abstract We introduce distinct production funtions for investment and consumption goods to the Solow (974) model of sustainable depletion of a nite oil stock. This endogenizes the price of investment goods and leads to two new scenarios of the using up of a nite stock of oil, while the the economy s consumption level remains unchanging. The new cases involve investment in durable capital increasing or decreasing as time passes. Introduction We introduce distinct production funtions for investment and consumption goods to the Solow (974) model of sustainable depletion of a nite oil stock. This endogenizes the price of investment goods and leads to a di erent scenario of using up of a nite stock of oil, while the economy s consumption level remains unchanging. Our model allows for the possibility of investment in produced capital declining or increasing over time, rather than remaining non-varying, as in Solow (974). Our model has then two, constant returns to scale, production functions q C = f(k C ; R C ); () and q I = g(k K C ; R R C ): (2) for q C and q I current output of consumption and investment goods respectively. Inputs K C and R C are services of produced capital and oil input respectively to the consumption goods sector. There are two zero pro t relations

q C = rk C + vr C (3) p I q I = r[k K C ] + v[r R C ]: (4) The price of consumption goods is set at unity. r is the rental price per unit of capital K and v is the price for a unit of oil, R: We have K I = K K C and R I = R R C. There are static e ciency conditions for input use in f RC f KC f RC = r v and g K I g RI = r v : (5) is the derivative of f(:) with respect to R C : In solving the model, we impose the condition that q C is non-varying (q C = q C ), and current oil rents fund current investment. q C non-varying is taken as a condition for the economy to be on a sustainability trajectory and funding current investment with current exhaustible resource rents is taken as a savings "behavior". That is vr = p I q I : (6) Our model is seven equations in q I ; p I ; K C ; R C ; r; v and R: Initial "endowments" are K and S 0 : As time passes from date t to t +, K t increases by q I and S t decreases by R: We solved the 7 equation system for a sequence of dates with Matlab with each sector having a constant returns to scale, Cobb-Douglas production function. We set p C = q C = and q C = [K C ] 0:8 [R C ] 0:2 and q I = [K K C ] 0:3 [R R R ] 0:7 : The initial value of K is 5.0. K is increased across dates by q I : See Table for numerical results. Table qi p I K C R C r v v R t 0 0.0373 7.8687 4.0000 0.0039 0.2000 5.2000 5.773 0.030 t 0.0366 8.2038 4.0298 0.0038 0.985 52.7450 53.3202 0.026 t 2 0.0360 8.5363 4.059 0.0037 0.97 54.2947 54.8720 0.023 t 3 0.0353 8.8668 4.0879 0.0036 0.957 55.852 56.434 0.09 t 4 0.0347 9.944 4.62 0.0035 0.944 57.45 57.9928 0.06 t 5 0.0342 9.596 4.439 0.0034 0.93 58.9760 59.5593 0.03 2

K and p I are increasing. q I and R are decreasing. p I cannot move separately from p C in the Solow formulation; nor could q C vary over time. v is the current value of v increased at the current interest rate. Observe that the actual v rises at somewhat more than the currently projected v because we are working in discrete rather than continuous time. The solution is "rotating" along the isoquant for q C and the price of a unit of exhaustible resources is "high" relative to the rental rate, r for a unit of capital. Also the solution involves shifting down at each date to a "lower" isoquant for q I : We turn to an example with q I increasing over time. We set p I = q C = and now q C = [K C ] 0:4 [R C ] 0:6 and q I = [K K C ] 0:9 [R R R ] 0: : The initial value of K is 5.0. K is increased across dates by q I : q C = p C = : Results of our Matlab solving are in Table 2. Table 2 qi p I K C R C r v v R t 0 2.0602 0.3236 2.0000 0.6300 0.2000 0.9524.54 0.7000 t 2.7466 0.2427 2.824 0.5005 0.46.988.8983 0.556 t 2 3.67 0.846 3.9227 0.4020 0.020.4924 2.368 0.4467 t 3 4.6902 0.42 5.3674 0.3262 0.0745.8393 2.8037 0.3625 t 4 6.022 0.07 7.2435 0.267 0.0552 2.2462 3.3665 0.2968 t 5 7.6484 0.0872 9.6520 0.2206 0.044 2.7200 4.032 0.245 t 6 9.6209 0.0693 2.73 0.836 0.035 3.2680 4.7520 0.2040 The above system of equations can be solved without explicit prices as four equations in four unknowns: q I ; K C ; R C ; and R: Result : The above system of equations implies the Hotelling Rule, _g RI g RI = i for i = r=p I = g KI : The demonstration involves taking the time-derivative of q C and of q I = Rg RI : When we di erentiate q I = Rg RI we get g KI [ _ K _ K C ] + g RI [ _ R _R C ] _Rg RI R _g RI = 0: We use the static e ciency condition g K I g RI this to g RI f f K C f RC _K C + R _ C g + K _ g KI = f K C f RC and _ K = q I = Rg RI to reduce _g RI g RI = 0: 3

The term in braces equals zero because _q C = 0: Hence _g R I g RI = g KI ; which is the familiar Hotelling Rule (exhaustible resource price rises at the rate of interest). With Cobb-Douglas production functions, q C = [K C ] ac [R C ] ac and q I = [K K C ] ai [R R R ] ai ; our four equation system allows us to express q I in terms of K and other parameters, excluding S: We start with Rg RI for Cobb-Douglas, we get = q I and R C = ai R: (7) Since q C is a parameter, we have from q C = [K C ] ac [R C ] ac =( ac) q C R C = (K C ) ac : (8) Input-use e ciency yields ac ai RC = R ac ai K C K R C : K C (9) These three equations allow us to express R R C and K K C in terms of K and parameters of the production functions. Thus using q I = _ K = (K K C ) ai (R R C ) ai we get ai ai _K = ( ac) ai qc ai (ac) ac ai ac K [ ai ac ac ] : (0) We observe below that K(t) is increasing over time. Equation (0) suggests h i ai ac then three possibilities: () ac = 0 and K _ is unchanging (Solow (974), h i ai ac (2) ac > 0 and K and K _ are each increasing as time passes, and h i ai ac (3) < 0; K is increasing and K _ is decreasing as time passes. below. ac) ac This last case was illustrated in Table above. More on these possibilities Result 2: Equation (0) solves as K [ h ai ac] t = A [t + C] for A = ai ai ai ai qc [ ai ac] (ac) and C a positive constant de ned by the en- ac dowment K 0 : ai ac i ( Quasi-arithmetic growth in K(t) was central to the solution of the Solow Model with "extra savings" and endogenous population growth (Asheim, Buchholtz, Hartwick, Mitra and Withagen (2007)). 4

h Obviously, K(t) is increasing in time since ai ac i > 0: Our solution implies that _ K=K = B=(t+C) for B > 0: Hence _ K=K! 0 as time passes. This latter behavior appears in the Solow formulation also. Consider now the niteness of oil use over in nite time. We know that R = =( ac) q C ai (ac K) ac : Hence convergence turns on the niteness of R 0 Rdt or the niteness of ai qc (ac) ac =( ac) Z 0 K ac=( dt ac) for K = A [ ac ai ] [t + C] [ ac ai ] : That is, we are concerned about the niteness of The integral for Z = ai qc =( ac) [A] ac=( Z dt: () [t + C] Z (ac) ac ai) 0 ac ( ai) : (2) Z T 0 [t + C] Z dt = Z + [t + C]( Z+) j T 0 For T! ; we get = ( Z )f(t + C)( Z+) (C) ( Z+) g: ( Z )f(c)( Z ) g for Z > 0 or ac ( ai) > 0: Hence (Result 3) for the limiting value of the integral to be nite and positive, we require that ai < ac: If one takes this result to the expression in (), one observes that this condition is one on terms in the "sum" R 0 that get smaller relatively quickly as time passes (that is, ac=( We return to _ K = ( ac) ai ai ai ai ac [t+c] ac=( ai) dt ai) > ). ai ai qc ac (ac) K ac must be greater than unity. The case of [ ai ac ac ] : We know that niteness of oil use implies that _K declining is particularly novel: output from the investment goods sector is shrinking to zero in the limit while the use of oil in the consumption goods sector 5

is also declining to zero in the limit. One presumes that the level of sustainable consumption q C is very small for this new case. See Table for some solved time-steps in a trajectory for this case. 6

APPENDIX: function f=stol(x) ac=0.4;ai=0.9; K=5;qc=.0; % de ne terms needed in solving... pc=; v=x(); qi=x(2); pi=x(3); Kc=x(4); % Nc=x(5); Rc=x(5); r=x(6); Ki=K-Kc; % Ni=Nb-Nc; v=x(7); R=x(8); Ri=R-Rc; % f()=qc-(kc^ac*rc^(-ac)); f(2)=qi-(ki^ai*ri^(-ai)); % f(3)=ac*rc*v-(-ac)*kc*r; f(4)=ai*ri*v-(-ai)*ki*r; f(5)=pc*qc-(r*kc+v*rc); f(6)=pi*qi-(r*ki+v*ri); f(7)=v*r-pi*qi; % hotelling... f(8)=v-(+(r/pi))*v; % Base has K=5.0 and qc=0.872... outputs R=0.5 and qi=.992... then re-do % K+qi and get new shrinking R under qc CONSTANT... i=r/pi... %... Hotelling relations...pr and pr for approx i % motion (discrete time) 7

% 3.9843, qi=.992, pi=0.2735, Kc=2.0000, Nc=0.4500, r=0.634, v=.0896, R= 0.5000, pr=6.3654 % 6.5888 2.6343 0.2068 2.7968 0.3598 0.69.3626 R= 0.3998 0.324 % 0.6437 3.4386 0.584 3.8506 0.2908 0.0849.6864 R=0.323 6.3467 % 6.829 4.4353 0.228 5.2260 0.2372 0.0625 2.0672 R=0.2635 25.3988 % 26.0895 5.6585 0.0963 7.000 0.952 0.0467 2.59 R= 0.269 38.743 % 39.766 7.465 0.0762 9.2635 0.69 0.0353 3.0277 R= 0.799 58.006 % 59.4533 8.949 0.0609 2.22 0.354 0.0270 3.6223 R= 0.504 85.7666 % 87.625.098 0.049 5.6989 0.39 0.0208 4.3037 R=0.266 24.7662 % 27.2784 3.6484 0.0399 20.356 0.0965 0.062 5.0805 R= 0.072 79.0420 % IRR mode... very small R (=0.5) % 0.872.992 0.2735 2.0000 0.4500 0.634.0896 % same with added pr % 0.872.992 0.2735 2.0000 0.4500 0.634.0896 pr=3.9842 % pr exog, pr=3.9842, and R endog % 0.872.992 0.2735 2.0000 0.4500 0.634.0896 R= 0.5000 % new pr as pr, comes out % 0.872.992 0.2735 2.0000 0.4500 0.634.0896 0.5000 pr=6.3652 % 0.5980.890 0.208 2.0000 0.2674 0.96.349 0.297 pr= 9.9762 % 0.4432.7989 0.642 2.0000 0.623 0.0886.6385 0.803 pr= 5.360 %... % ac=0.2, s=.9 % 0.873, 6.7922, pi=.0830, 0.3448,.04, r= 0.4740, v= 0.6448 % ac=0.25... s=.9 % 0.7809 6.7909.0350 0.4237 0.9574 0.4607 0.6 % ac=.8, ai=.9, s=.9 % 0.5820 4.7749.0969 0.4494.6364.0359 0.07 ******** Asheim, Geir, Wolfgang Buchholz, John Hartwick, Tapan Mitra, Cees Withagen (2007) "Constant Savings rates and quasi-arithmetic Population Growth under Exhaustible Resource constraints" Journal of Environmental Economics and Management, 64, pp. 23-229. 8

Solow, Robert M. (974) "Intergenerational Equity and Exhaustible Resources", The Review of Economic Studies, Vol. 4, pp. 29-45. 9

Stollery Model with Public Goods Pollution in the Two Sectors 2 Introduction We emphasize that a pollutant such as a greenhouse gas is a public-good commodity, entering here equally in the production functions of all producers. This implies that the allocation analysis for an economy with global warming must keep public-good charges front and center. We make this case with an extension of the one-sector Stollery (996) "growth" model, a model descending directly from the Solow (974) sustainability model with oil extraction and stock depletion. Firms in our consumption goods and investment goods sectors each end up paying a Pigovian tax (a "price" for temperature) in their respective current zero pro t relations, a tax which is a public goods charge of the Samuelson (954) type (our public good is actually a public bad and is an intermediate good rather than a nal good 2 ). These rm-speci c charges sum to the public good "price" for temperature which appears in the zero arbitrage pro t condition for extractors. In brief, we disaggregate the Stollery model of global warming into two distinct sectors, pay close attention to the public goods nature of pigovian charges, and take up the possibility of solving the model for the case of each production function of the Cobb-Douglas form. 3 The Analysis At each date the economy is producing q C of consumption goods and q I of investment goods, each with a distinct constant returns to scale production function. The inputs are capital K and oil R; and inverse-temperature Z: That is, Z is =T for temperature T and T re ects cumulative extraction of oil. For example, T might take the form [S 0 S(t)]: Each period involves more oil extraction, R and thus a rising temperature since cumulative extraction increases by R: Temperature is the direct "bad" imposing a current cost on producers. Population is assumed to be unchanging and we leave labor and population out of the production functions. 2 There is a large literature on intermediate public good inputs. See for example..., Sandmo, MacMillan, Manning, Markusen, etc. 0

Our model has then two production functions q C = f(k C ; R C ; Z); (3) and q I = g(k K C ; R R C ; Z): (4) The static e ciency condition for use of K and R is f KC f RC = g K I g RI (5) for K I = K K C and R I = R R C :In solving the model, we impose the condition that q C is non-varying (q C = q C ), and current oil rents fund current investment. That is Rg RI = q I : (6) Our model is four equations in q I ; K C ; R C and R: Initial "endowments" are K and S 0 : As time passes from date t to t +, K t increases by q I and S t decreases by R: We consider the analysis for the moment with Z as a generic externality to producers in each of the consumption and investment goods sectors. We would of course like our externality to impinge on each rm in the economy but we cannot break out distinct rms, given contant returns to scale at the level of the industry. Hence from a public goods standpoint we have two distinct competitive rms, one doing consumption goods and one doing investment goods. Result : The above system implies the amended Hotelling Rule, " # _g RI _ Z ZgZ = g KI + + Zf Z : (7) g RI Z Rg RC Rf RC The demonstration involves taking the time-derivative of (6) and then using _q C = 0: When we di erentiate (6) we get g KI [ _ K _ K C ] + g RI [ _ R _R C ] g Z _S _Rg RI R _g RI = 0:

We use R = _ S; the static e ciency conditions (5) and (??), and _ K = q I = Rg RI to reduce this to f g R I f KC f RC _K C + g K I f RC f KC _R C g + g Z _Z + K _ g KI _g RI g RI = 0: We add and subtract g R I f Z _Z f RC to get f g R I f KC f RC _K C + g K I f RC f KC _R C + g R I f Z _Z g+ g R I f Z _Z +g Z _Z + K f RC f _ g KI RC _g RI g RI = 0: The term in braces equals zero because _q C = 0: The remaining terms reduce to (7). h i h Z _ The term ZgZ Z Rg RC + Zf Z represents the "distortion" from the familiar Hotelling Rule (exhaustible resource price rises at the rate of interest). This Rf RC i negative term is a pigovian tax on extraction (a source tax) that induces extractors to slow extraction because a larger R induces currently a larger increase in temperature. Observe that each of Zg Z and Zf Z are own Pigovian taxes associated with the investment and consumption goods sectors respectively. They sum in (7) because they are charges for a public bad, namely temperature, Z: We can consider the case of prices for our system, with q C with price unity and q I with price p I : Then we would have f RC = v = p I g RI and f KC = r = p I g KI : Our adjusted Hotelling Rule would then be " # _g RI _ ZfZ Z + Zp I g Z = g KI + g RI Z vr Zf Z +Zp I g Z are the public goods charges or total tax revenue for "public good", Z: Our Hotelling expression in (7) is a generic zero arbitrage pro t condition for extractors and is derived in Appendix in a fairly generic growth model (the Dasgupta-Heal (974) model). Hence what we have essentially is that investing exhaustible resource rents implies (7), a dynamic e ciency condition. Some : 2

may nd the "inverse" expression more congenial, namely unchanging consumption and e ciency conditions, including the dynamic "adjusted" Hotelling rule, implies the savings investment rule: invest current resource rents in current capital accumulation. Given the expression in (7), this proposition is easy to demonstrate (it makes use of procedures in our derivation above). 4 Numerical Investigation We turn to solving the system for the case of production functions of the Cobb- Douglas form. We have q C = [K C ] ac [R C ] acr ( ac acr) [Z] and q I = [K K C ] ai [R R C ] air [Z] ( ai air) : Since q C is unchanging, we have The static e ciency condition gives us =acr q C R C = : [K C ] ac [Z]( ac acr) ac air acr ai R C = R R C : K C K K C The investment condition Rg RI = q I becomes R C = ( air)r: This allows us to reduce the e ciency condtion to ac ( air) K K c = (acr ai) + (ac ( air)) : We proceed to substitute into _ K = [K K C ] ai [R R C ] air [Z] ( ai air) : First, we get _K = K ac ( air) (acr ai) + (ac ( ( ac air (K K C ) ai acr K C air)) q C [K C ] ac [Z]( ac acr) ai ( ai air) fzg =acr ) air : 3

Some additional substitution yields _K = C K (aiacr acair)=acr Z [( ai air)(acr) ( ac acr)(air)]=acr] (8) C = for acr ai (acr ai) + (ac ( ( ai air acr ai + ( air)) ( air) ac( air) ac air) ac acr q [=acr] C ) air : Both K and Z on the right side enter with an exponent (i.e. relatively simply). Note that for the special case of each sector having the same production function (Stollery (996)) we have _ K a constant, as in the original Solow model. For this case, the temperature externality is "signi cant" only in the _ S equation, below. Stollery was aware that his formulation involved _ K a constant (K(t) = A+Ct; with A a positive constant) and he exploited this property in obtaining an explicit solution of this system. See below for details. The companion equation to our two equation system is _S = R = ( air) R C 2 = 4 ( air) [ q C ac( air)k (acrai)+(ac( air)) ]ac [Z]( ac acr) 3 5 =acr = C2 K ac=acr Z ( ac acr)=acr (9) C2 = for 2 4 ( air) [ q C ac( air) (acrai)+(ac( air)) ]ac 3 5 =acr : We now make Z an explicit function of S and consider our two simultaneous non-linear di erential equations in K and S: Two possibilities are T (t) = T 0 exp( S(t)) ( and T 0 positive; Stollery (996), and T (t) = [S 0 S(t)] with Z = =T: We use the Stollery form in our calculations reported below. Considerable information can be gleaned from numerical solving with parameters of reasonable values. () Identical production functions for q C and q I : (Stollery (996)) 4

Equation (9) becomes _S = [A + C t] ac=acr C2 [=(T 0 exp( S(t)))] ( ac acr)=acr Numerical investigation is not trivial for this case. Well-behavedness requires that the exponent for Z in equation (9) be small in absolute value and the value of acr be larger than ac: Well-behavedness involves solutions with S(t) convex in t and asymptotic to the y axis. Solving numerically with Maple software yielded such solutions. (2) We then took up cases with the exponent for S in equation (8) equal to zero. Such cases can be described as those following Hartwick (203), with a temperature externality added to each production function. We simply "translated" our system in (8) and (9) into discrete time and solved for time paths of K and S: We chose K 0 = 0 and S 0 = 2 for each case. We chose parameter values for the production functions that left the exponent on K in equation (8) close to zero (the exponent for S was set at zero). (a) With the exponent on K > 0 in equation (8), S(t) declined with the absolute value of S declining, and K(t) increased with K increasing. (b) With the exponent on K < 0 in equation (8), S(t) declined with the absolute value of S declining, and K(t) increased with K decreasing. (3) We then took up cases with the exponent for K in equation (8) equal to zero. (a) With the exponent on S > 0 in equation (8), S(t) declined with the absolute value of S declining, and K(t) increased with K decreasing. (b) With the exponent on S < 0 in equation (8), S(t) declined with the absolute value of S declining, and K(t) increased with K increasing. (4) With the exponent on Z in equation (8) negative and the exponent on K positive, we observed K increasing with K increasing and S declining with the absolute value of S declining. 5

APPENDIX : THE DASGUPTA-HEAL MODEL with two distinct Sectors The Dasgupta-Heal (974) model involves Ramsey saving (present value of the utility of consumption is maximized with a constant discount rate ) and oil use in current production, where current oil used is drawn from a nite stock, S: Population (the labor force) is taken to be unchaning. We simply restate their model and its necessary conditions for the case of consumption goods and investment goods each having a distinct, constant returns to scale production function. q C = f(k C ; R C ; Z) q I = g(k K C ; R R C ; Z) Z = [S 0 S t ];... temperature, Z... [S 0 S t ] is cumulative extraction. q I = _ K and R = _ S: The current-value Hamiltonian is Necessary conditions are H = u(q C ) + [q I ] R for q I = _ K and R = _ S: @H @K C = 0 ) u qc f KC = g KI ; for p I = =u qc and r = f KC = p I (eleven) g KI ; @H @R C = 0 ) u qc f RC = g RI ; for v = f RC = p I g RI (thirteen) @H @R = 0 ) g R I = ; (fourteen) @H @K = _ or g KI = _ ; ( fteen) @H @S = _ or [u qc f Z + g Z ] dz ds = _ : (sixteen) Combining (5) and (6) yields _g RI g RI = g KI + [ Zf Z Rf RC + Zg Z ] dz R Rg RI ds Z (20) This is the amended Hotelling Rule. The form of (20) di ers slightly from the corresponding term in the text because in the above planning formulation, the 6

planner takes account explicitly of the dependence of the externality Z on stock S whereas in the formulation in the text, rms react to the externality without linking the externality to current stock size S: In the text, rms are treating Z as a parameter whereas above Z is being treated as a function of stock size S: 7

APPENDICES WITH BACK-UP MATERIAL... example with "arbitrary" initial values (K=0, S=2) and parameters... c=;c2=;ac=.3;acr=.5;ai=.4;air=.45;t0=.0;mu=0.5; % (ai*acr-ac*air)>0 in rst equation... % ((-ai-air)*(acr)-(-ac-acr)*air)<0 in rst equation % delt K (pos) is getting larger and delt S (neg) is getting smaller. % K=0, S=2 %.267.9772 % 2.2696.9557 % 3.4275.9354 % 4.5994.96 % 5.7845.8976 % 6.9820.8799 % 8.92.8629 % 9.46.8465 % 20.6426.8307 % 2.8838.854 % 23.348.8006 % 24.395.7863 % 25.6644.7724 % 26.9424.7589 % 28.2287.7457 % 29.523.7328 % 30.8253.7202 % 32.35.7079 % 33.4523.6959 % 34.7766.6842 % 36.078.6727 % 37.4458.664 % 38.7903.6504 % 40.42.6396 % 4.4984.6290 % 42.867.686 8

% 44.2309.6083 % 45.6060.5982 % 46.9868.5883 K increasing and S decreasing in abs value ///////// example with exponent of K in rst equation set at zero (start K=0,S=2) c=;c2=;ac=.45;acr=.5;ai=.4;air=2/4.5;t0=.0;mu=0.5; % (ai*acr-ac*air)=0 in rst equation...(yields declining K) % expon on S : ((-ai-air)*(acr)-(-ac-acr)*air)=(.07777-.02222)>0 in rst equation % K=0; S=2 %.9477.9309 % 3.8880.878 % 5.829.820 % 7.7503.7740 % 9.6737.7323 % 2.5927.6942 % 23.5076.659 % 25.488.6265 % 27.3265.596 % 29.230.5676 % 3.325.5407 % 33.032.553 % 34.9272.492 % 36.8206.4682 % 38.76.4462 % 40.6003.4252 % 42.4868.405 % 44.372.3857 % 46.2536.367 % 48.340.349 % 50.026.337 % 5.8893.349 9

% 53.7643.2987 % 55.6376.2830 % 57.5093.2677 % 59.3794.2529 K decreasing (toward zero?) and S decreasing in abs value. //////// Other example with exponent on K at zero (reverse coe cients from above case) c=;c2=;ac=.4;acr=2/4.5;ai=.45;air=.5;t0=.0;mu=0.5; % (ai*acr-ac*air)=0 in rst equation... % expon on S : ((-ai-air)*(acr)-(-ac-acr)*air)=.02222-.07777<0 in rst equation % K=0; S=2 % 0.4724.9846 % 0.9452.9698 %.485.9555 %.8922.947 % 2.3663.9284 % 2.8408.955 % 3.357.9030 % 3.7909.8909 % 4.2665.879 % 4.7424.8677 % 5.287.8566 % 5.6953.8458 % delt K increasing and delt S decreasing in abs value (this delt K is di erent from previous example) ////////// expon on S at zero c=;c2=;acr=2/4.5;ac=.55555;t0=.0;mu=0.5;ai=.05;air=.5; K= 3.6873;S=.9904; K=x(); S=x(2); 20

Z=/(T0*exp(-mu*S)); f()=k-(k+c*k^((ai*acr-ac*air)/acr)*z^(((-ai-air)*(acr)-(-ac-acr)*air)/acr)); f(2)=s-(s-c2*k^(-ac/acr)*z^(-(-ac-acr)/acr)); % ((-ai-air)*(acr)-(-ac-acr)*air)=0 in rst eqn % ai*acr-ac*air=.02222-.077777<0 (expon on K) in rst equation % K=0; S=2 % 0.7499.9980 %.4930.9960 % 2.2300.994 % 2.963.9922 % 3.6873.9904 % 4.4083.9886 % delt K declining; absval delt S declining /////// expon on S=0 c=;c2=;t0=.0;mu=0.5; acr=.5;ac=.05;ai=.55555;air=2/4.5; K= 7.986;S=.9789; K=x(); S=x(2); Z=/(T0*exp(-mu*S)); f()=k-(k+c*k^((ai*acr-ac*air)/acr)*z^(((-ai-air)*(acr)-(-ac-acr)*air)/acr)); f(2)=s-(s-c2*k^(-ac/acr)*z^(-(-ac-acr)/acr)); % ((-ai-air)*(acr)-(-ac-acr)*air)=0 in rst eqn % ai*acr-ac*air=.07777-.022222>0 (K exponent) in rst equation % K=0; S=2 %.296.9964 % 2.6007.9928 % 3.9259.9893 % 5.2659.9858 % 6.696.9823 % 7.986.9789 % 9.3647.9755 2

... delt K increasing; absval delt S decreasing ///////////// 22